Shafer [12] showed that this approximation is an interesting alternative to Pad ...... [9] Khéira S. (2005), Approximants de Hermite-Pad dé erminants ...
!
"! #
$ % !& ' ' () ' ( * + ' ,
! " #
! " # $ !% ! & $ ' ' ($ ' # % % ) % * % ' $ ' + " % & ' !# $, ( $ - . ! "- ( % . % % % % $ $ $ - - - - / / $$$ 0 1"1"#23."
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@
1
f sin 2 T f 2 fT2 f\2 2 dT d\ is the surface energy of the drop,
0 0
7
O 5
SS /2
³ ³f
5
sin 3 T dT d\ is the kinetic energy of the drop considered as a rigid body, and
0 0
V is the scaled drop volume. k is considered as Lagrange multiplier introduced with the volume constraint (2.3). The subscript denotes partial differentiation, i.e. f T
wf , f\ wT
wf . w\
In fact, the first variation of ( with respect to all shapes is:
į( K į6 K į7 K į9 K , with
G6 K
G7 K
SS /2
f sin 2 T fK fTKT f\ K\ ° 2 2 2 2 sin K T f f f ® T \ ³ ³ sin 2 T f 2 fT2 f\2 0 0 ° ¯
SS /2
³ ³ OK f
4
sin 3 T dTd\ , GV K
0 0
SS /2
³ ³K f
2
½ ° ¾dT d\ ° ¿
sin T dTd\
0 0
where K T ,\ is the variation of the equilibrium shape f T ,\ subject to conditions (2.1-2). After some calculations, we get:
į( K
³ ³ >Of
SS /2
2
@
sin 2 T H k K f 2 sin T dT d\
0 0
25
0 , for all K .
The standard argument of the calculus of variations shows that f is a stationary point of ( if and only if f is a solution of the Young-Laplace equation (2.5). Let us notice that the sphere :, given by the equation f(T\) = 1 is a solution of the problem. For O z 0 one can look for an axi-symmetrical solution. Therefore f is independent of \. In term of interface equation, the curvature H can be written:
H
ffTT 2 fT2 f 2
f
2
2 3/ 2
fT
fT cosT f sin T
S /2
The constraint of volume (2.3) reduces to:
(2.6)
sin T dT 1 0
(2.7)
1/ 2 f sin T f 2 fT2
³f
3
0
3. Method of solution Using the implicit function theorem (cf. appendix), one can show that there is a unique analytic solution in O. Therefore f(T) and k may be written as: f T
k
¦O ng n T
nt0 On hn nt0
¦
(2.8) (2.9)
The straightforward method for obtaining the successive governing equations to determine ( g n , hn ) is to substitute (2.8-2.9) into (2.5)-(2.7) and to collect the coefficients of like powers
ofO. Doing so, g n are solutions of the ordinary differential equations hierarchy (cf. appendix): Lg n
where
An g 0 , g1, , g n 1; h0 , hn 1 , with g 0 T 1 .
L
d
2
dT 2
sin T d 2 Id cosT dT
(2.10)
(2.11)
Id is the identity operator and An is a nonlinear differential operator. Equations (2.10) are subjected to condition (2.7) in order to calculate hn . Equations like (2.10) FDQ EH VROYHG DQDO\WLFDOO\ XVLQJ D EDVLV RI /HJHQGUH¶V SRO\QRPLDOV Pn x . One notice that only the even order Legendre polynomials are involved in the
26
Oc lies on the positive axis [28].Table 2.1 and 2.2 show the rapid convergence of the
dominant singularity Oc together with its corresponding critical exponent J c with gradual increase in the number of series coefficients utilized in the approximants. M
N
OcN
J cN
2
12
2.277088792
0.4662097707
3
15
2.273591366
0.5101738986
4
18
2.274504009
0.4936626763
5
21
2.274078652
0.4999538775
6
24
2.274086883
0.4998963588
7
27
2.274080602
0.4983599626
8
30
2.274084226
0.4999692598
Table2.1: Computations for nearest singularity of the aspect ratio e(O) showing the procedure rapid convergence M
N
OcN
J cN
2
12
2.279785204
0.4400114597
3
15
2.157214717
0.5101738986
4
18
2.274026098
0.5010615885
5
21
2.274086469
0.4994713197
6
24
2.274120573
0.4993306037
7
27
2.274086119
0.5000285734
8
30
2.274085206
0.4999782565
Table 2.2: Computations for nearest singularity of the reference pressure k(O) showing the procedure rapid convergence One can see in Table 2.1 and 2.2 that the direct calculation gives the estimation Oc | 2.2742, J c | 0.5.
28
By similar calculations, we proceed to the extraction of singularities of the equatorial and the polar radius. We notice that they have the same value of the aspect ratio singularity.
1
0.
0.
;ĂͿ
e(O
0.
0.
0
0.
1
O
1.
2
Figure 2.2. Aspect ratio versus O; (a) calculated, (b) expected form.
29
2.
12 10 8
e-1
6 4 2 0 -2 -4 -1.5
-1
-0.5
0
0.5
1
1.5
O
Figure2. 3. Plot of the function O O (e 1) It is possible to calculate the truncated series (2.24) defining e(O) by a division procedure. This function is plotted in figure 2.2a. The graph of e(O) (figure2.2a) is not compatible with the expected singularity (2.16). The aspect ratio should be of the form given in figure 2.2b. To analyse the paradox, the inverse function O O (e 1) is considered. Its series is given by inverting the series e(O) [28]. One can see in figure2.3 that this function has a horizontal tangent for the values (e - 1 = 0.7926,
O The maximum value of Ocorresponds to the value of Oc up to three digits, which confirms the existence of a singular point, see [55]. The results show that there is a maximum value of the Bond number Oc = 2.2742 above which there is no axially symmetric drop. This value is somewhat smaller than the approximate value (2.4544) obtained by Chandrasekhar [45] with elliptical integrals, but close to that found in [46] (2.3222) where the deformation of an axisymmetrical drop is computed 30
numerically. The results also show that as Oincreases, the drop becomes oblate, spreading out in the direction normal to the rotation axis and contracting in the direction of the rotation axis. At Oc , the drop shape changes from axisymmetric to two lobed, and just beyond that value to four lobed. 5. Conclusions Equilibrium shapes of rotating liquid drops are found using perturbation technique. A bifurcation study by analytic continuation of a power series is performed using a special Hermite-Padé approximants. The above computer-extended series solution method in conjunction with Hermite-Padé approximation technique is advocated as an effective tool to investigate several other parameter dependent nonlinear boundary value problems.
31
Chapter 3
Axisymmetric shape of gas bubbles in a uniform flow
32
1. Introduction The rupture of liquid drops by a gas flow is of great importance in engineering and technology, including spray painting, agriculture (drip irrigation, pesticide drift), atomization etc. In combustion theory, models of fuel droplets are used to describe droplet vaporization in a high-temperature environment in devices such as diesel engines, gas turbines, furnaces and rocket motors [57-59].The situation of a single drop subjected to aerodynamic forces due to slip velocity between the drop and the surrounding gas can also simulate a raindrop falling through the atmosphere [60]. Since the early 1950's Pany theoretical and experimental works has been performed on the study of the deformation of a liquid drop or gas bubble in irrotational flow. The analysis of the forms goes back to Davies and Taylor [61]. They studied air bubbles rising in nitro benzene or water through measurements of photographs carried out by spark photography. Furthermore they found an analytical expression relating the velocity of rise and the radius of curvature in the region of the vertex. The parameters that govern the movement of a rising or falling drop or bubble, and consequently characterize its shape, lead through a dimensional analysis to three independent groups of dimensionless numbers: Reynolds, Morton and Weber numbers. The Weber number represents the ratio between the forces tending to deform the bubble and those which tend to stabilize the interface. Hence, the Weber number indicates whether the kinetic or the surface tension energy is dominant. Theoretical analysis has been directed towards the study of the interaction of the surface tension and hydrodynamic pressure leading to unstable forms, i.e. the existence of a critical Weber number. This critical value corresponds to a subcritical bifurcation, which means that above this value there is no stable equilibrium position. In this case, due to the growth of capillary wave instabilities, the drop breaks up. This is one of the atomization processes; see [62]. The 33
accurate determination of this critical value is useful for the calibration of nozzles producing calibrated drops. In the following, we will briefly review some of the previous results and present some new along this line of research. Saffman [63] examined the rise of small air bubbles and presented both experiments and theory to characterize WKHEXEEOH¶VWUDMHFWRU\IURPa straight line vertical to a zigzag pathortoa spiral movement. In the planar zigzag motion, he determined the value of the Weber number at which the rectilinear motion becomes unstable. Hartunian and Sears [64] investigated both experimentally and theoretically the instability of small gas bubbles moving uniformly in various liquids. By assuming that the surface remains sphericaleven for a positive Weber number, they found a critical value of the Weber number DERYHZKLFKWKHEXEEOH¶VVKDSHEHFRPHVXQVWDEOH. Moore [65, 87] considered the rectilinear motion of an oblate ellipsoidal bubble and obtained a first approximation to the drag coefficient which is an essential factor to predict the trajectory of the bubble. El Sawi [66] examined the same problem using an appropriate approximation based on the virial method. He found that the moving air bubbles in the water are deformed and their forms can be approximated by an ellipsoid of revolution. Analytical expression relating Weber number and the aspect ratio of the bubble was derived, which led to an accurate and close value compared to that obtained by the previous two authors. He also extended his study, taking into account the effect of gravity. During the last decades, several investigations have been conducted on drop shapes and stability of this type of free boundary problem, see for instance [67-72] and the references therein.
34
The main purpose of this paper is the investigation of axisymmetric shapes of a deformable bubble in a uniform flow. We use a domain perturbation method which consists in taking as principal unknown a transformation field from the initial known position of the bubble to the unknown position depending on a positive parameter corresponding here to the Weber number. Velocity and pressure fields relating to flow and the shape of the bubble are determined as a power series in terms of the Weber number. Bifurcation and turning points are located by applying a special type of Hermite-Padé approximation. Traditionally this method has been used to determine the location and the nature of the singularity of a function from its power series expansion, as well as its evaluation on its branch cuts. This technique has been expanded by several investigators to find the singularities related to different physical problems. In the following sections, the problem is formulated, analyzed and discussed. 2. Governing equations We consider a steady flow caused by rectilinear motion of a bubble or drop with constant velocity V. Viscosity and gravity are both neglected. We denote by : the closed domain occupied by the fluid, by w: the boundary of :and by n its outward unit normal vector. The surrounding fluid occupies the domain :e
IR 3 \ : . The situation is portrayed in Figure 3.1.
The shape of the bubble is governed by the balance between the dynamic of the pressure distribution and the surface tension. The problem will be set in a reference frame Oxyz with the origin attached to the centre bubble. We assume without loss of generality, that V is parallel to the coordinate axis Oz (cf. figure 3.1).
35
z :
e
w:
:
Vf y
O x
Figure 3.1. Schematic profile of the drop. The fluid velocity at infinity is then given by Vf
Vfe z , where e z is a unit vector along axis
z. We assume throughout that the flow is incompressible and irrotational. The latter condition implies the existence of a potential function u due to the presence of the bubble, such that the Vf u , with u o 0, and u o 0 at infinity. The
velocity field may be written as v
incompressibility condition v 0 and the slip condition at the boundary of the bubble gives the following governing equations for the fluid flow in the unknown domain :. 'u ° ® wu ° ¯ wn
0
in :e
(3.1.1)
Vf .n
on w:
(3.1.2)
The momentum equations reduce to the Bernoulli equation:
U
v2 p 2
U
Vf2 pf 2
(3.2)
where p is the pressure field, pf is the ambient fluid pressure and Uis the volume density. We assume that the internal bubble pressure p0 inside the drop is constant. This assumption is consistent with the bubble experiments of Walters and Davidson [73, 74] showing that the bubble volume remains approximately constant during its evolution. In the presence of surface tension forces, there exists a discontinuity in pressure across the bubble interface with the fluid pressure at the bubble boundary given by the Laplace relation p is the surface tension and C the mean curvature of the surface w:. 36
p0 V C , where V
Equation (3.2) can now be written as: ª1 ¬2
º ¼
U « u u.Vf » V C 2
pf p0
'p
(3.3)
In the following, we establish a variational approach to the problem and show that the positions of relative equilibrium are obtained by finding stationary values of the energy of the system and vice versa. In fact the surface and kinetic energy are given by:
EV : V
³ dS
U
, Ek :
2
w:
³v
:
2
dx
e
The Hamiltonian which corresponds to the total energy of the system is given by:
H : EV : Ek : We expect that H : must be stationary for small variations about equilibrium solution, i.e. that G H : 0 .We can see that the kinetic energy is unbounded. Let us consider a sequence of bounded domains :m
:e Bm , where Bm is an open ball centered at a point of : of
radius m large enough to contain the domain : (See Figure 3.2). As m goes to infinity, the sequence : m converges to :e .
:
e
:m w: m
Figure 3.2. Schematic representation of : m
37
*m
Let us define: Ekm :
³
U
:m :
v2 dx U 2
³
:m :
U
Vf2 dx 2
Then the kinetic energy can be defined as: Ek :
2
³
:m :
u
2
2u.Vf dx
lim Ekm : .
mof
Using Green's formula and equations (3.1), we obtain: Ekm :
· · U§ wu wu ¨ ³ u dS 2 ³ uVf . ndS ¸ ¨ ³ u dS 2 ³ uVf . ndS ¸ ¸ ¸ 2¨ wn wn 2¨ w: *m © w: ¹ © *m ¹
U§
(3.4)
By taking the limit as mo observing that Vf . n is a bounded value and using the conditions u o 0, u o 0 at infinity, we see that the second term on the right hand side of the expression (3.4) tends to zero as m o . Thus
Ek :
U 2
wu
³ u wn dS
w:
U 2
³ uVf .ndS
w:
The first variation of the surface energy is obtained by noting the following result (See for example reference [75]):
G EV : V
³ C Ș. Q dS , for anydisplacement field Kwith ǤK= 0. On the other hand, by
w:
a straightforward calculation, we obtain:
G Ekm :
U 2
³
w : m :
u
2
2u .Vf Ș. Q dS 2
³
G u .Y d[
: m :
Where G u is the first variation of the potential u given by: G u
u . Ș . We apply again
Green's formula to obtain:
³
:m :
G u .v dx
wu
³ G u Vf .ndS ³ G u . wn dS
*m
*m
38
Since
wu tends to 0 as mo and G u is bounded, we get lim wn m of
choosing K with compact support satisfying supp Ș *m lim
m of
³ G u Vf . ndS
³ ¨© 2 u §U
Thus G H :
lim
m of
*m
w:
2
wu
³ G u . wn dS
0 . Now, by
*m
, we obtain:
³ u . Ș 9f . Q dS
0.
*m
· 2u .Vf V C ¸ Ș. QdS ¹
2 ª1 º we deduce the expression: U « u u.Vf » V C ¬2 ¼
0 , Ș such that ǤK= 0, from which
constant , which looks similar to (3.3).
In the following, we are looking for a dimensionless formulation of the problem. For Vf
0 the equilibrium shape is a sphere : 0 ofradius R. It is then convenient to use R as
a length scale and to introduce the following dimensionless variables: u Vf Ru , r C
Rr ,
RC . Then, equations (3.1) and (3.3) become:
'u ° ® wu ° ¯ wn
ª1 ¬2
e
e z .n
on w:
º ¼
O « u u . e z » C where O
R UVf2
V
3.5a 3.5b
in :
0
2
k
(3.6)
is the Weber number which is the ratio of inertial forces to interfacial ones.
The reference pressure k
R'p
V
is an unknown constant depending on O, and will be
determined by the volume constraint: V
4 3 S R V0 . 3
(3.7)
Bar symbols are henceforth omitted, without fear of confusion.
39
If O= 0 (i.e. Vf
0 ), the equilibrium shape is a spherical domain : 0 taken as a fixed
reference domain. If O z 0 , the equilibrium configuration is the unknown domain :. So O will be considered as an expansion parameter for this problem, as will be explained further below.
3. Principle of domain perturbation method )RU VRPH IUHH ERXQGDU\ YDOXH SUREOHPV LQ IOXLG PHFKDQLFV D SDUDPHWHU XVXDOO\ QRQ GLPHQVLRQDOL]HGVD\ HLVLQYROYHG$OOTXDQWLWLHVLHWKHGRPDLQ:HRFFXSLHGE\WKHIOXLGDQGWKH PHFKDQLFDOTXDQWLWLHV YHORFLW\SUHVVXUHDUHJHQHUDOO\GHQRWHGE\XHGHSHQGLQJRQH9HU\RIWHQ IRUDSDUWLFXODUYDOXHRIWKLVSDUDPHWHUVD\ H WKHVROXWLRQ :XLVNQRZQ Such problems may be solved by using domain perturbation technique; a method initiated by Hadamard [76], developed by Joseph and co-workers [77-79], and has been successfully used by many authors in various fields [1, 25, 80, 81]. Zolésio [53] extended the boundary variation technique of Hadamard via the speed method and its developments (see [25]). This method can be used to find solutions to free boundary problems like the one proposed here, in which the domain has to be determined. To explain the principle behind the domain perturbation method, we begin by noting that the problem defined on :H is mapped on to a fixed reference domain : which is then perturbed to yield the actual geometry. Next we will regardHas a perturbation parameter and naturally seek solutions as perturbation series in H. 4. Method of solution 7KH GRPDLQ SHUWXUEDWLRQ PHWKRG FRQVLVWV LQ FRQVLGHULQJ DV SULQFLSDO XQNQRZQ WKH WUDQVIRUPDWLRQ ILHOG [ 7 ; O IURP WKH LQLWLDO NQRZQ SRVLWLRQ : FRQVLGHUHG DV D UHIHUHQFH SRVLWLRQ WR WKH XQNQRZQSRVLWLRQ,QIDFWWKLVILHOGGHSHQGVRQO
40
x = T(X͕O
X
:Ğ
x Ž
Ž :
:0
)LJXUH/DJUDQJLDQUHSUHVHQWDWLRQRIWKHGRPDLQ Let us describe briefly the domain perturbation method, cited above. Once equations (3.5-3.6) are transported back to the sphere, we get a new system of partial differential equations say F(T, X, O) = 0 written on the known domain :0. The set of all transformations x = T(X, O) can be considered as a one parameter group of transformations, and so we can say that x= (Id +
OT1 + O2T2 « X) where Id is the identity. A spherical coordinate system (r,T, \ has to be assumed, with the origin at the centre of the drop and with the z axis as the rotation axis as shown in Figure 2.1. Let T the transformation field from the initial equilibrium position :0, i.e. for O
0 to the
actual equilibrium position :i.e. for O z 0 . The form of the transformation field is chosen: T
rf (T ,\ , O )er
(3.8)
For axisymmetrical shapes, the radial shape function f and the potential depend only on the azimuthal angle T. Therefore the interface shape function is given by: r = f(T,O). Now under these considerations, the potential u can be written as follows: u( x, y, z )
v( r , T , O )
If V is changed to - V the configuration is unchanged, then by symmetry: f (S T , O )
f (T ,O )
(3.9)
The equations transported back to the reference configuration :0are given by:
41
1 t 1 ° detTc Tc Tc v 0 ° ° t 1 t 1 t c 1 T e z t Tc1n ® Tc v Tc e z ° 2 ° O ª 1 t Tc1v t Tc1v t Tc1e º C z» « °¯ ¬ 2 ¼
for r ! 1 for r 1 k
(3.10.1-3)
for r 1
1 t Where T c , T c , T c denote respectively the Jacobian, the inverse and the transpose of the
Jacobian T c . The reference pressure k is obtained by constraining the drop volume to have a constant value V0 as in (3.7): S /2
³
f 3 sin T dT
(3.11)
1
0
Equation (3.10.1) can be written as: F1r
2 1 cos T F1 F2T F2 r r r sin T
forr> 1
0
(3.12)
where
§ f2· f F1 r ,T ¨ f T ¸ vr T vT f r © ¹
,
F2 r ,T
f vT fT vr r
The subscript notation denotes partial differentiation. The boundary condition (3.10.2) becomes:
f
2
fT2 v r ff T vT
f
2
fT2 cosT ff T sinT
for r = 1,
(3.13)
While relation (3.10.3) takes the following form:
1
ª
¯2
¬
O ® ¬ªvr2 F32 ¼º vr cos T F3 «sin T with: F3 r ,T vT
fT vr º ½ 2 ¾ C k f f »¼ ¿
0
for r 1 ,
fT v r . f
In term of interface equation, the curvature C can be written as follows:
42
(3.14)
C
ff 2 fT2 f 2 f cos T f TT T 3/ 2 2 2 f fT f sin T f 2
sin T fT2
1/ 2
.
The advantage of this procedure is that the equations and boundary conditions are formulated on a fixed reference domain and consequently, explicit equations estimating the shape f are obtained. The major disadvantage is that these equations are a little more complicated. Notice that for O
0 there is a solution: f(T, 0) = 1, and v(r, T, 0). Using the Nash-Moser
implicit function theorem [80], we can assert the existence of a unique solution depending only on r, T and analytic inO The solution of the problem v, k and f will be then sought as series of O: n °v r , T , O ¦ O wn r ,T ° nt0 °° f T , O ® ¦ O n gn T ° nt0 ° n k O ¦ hn ° °¯ nt0
(3.15)
One has to substitute relations (3.15) into the governing equations (3.11-3.14), collect the coefficients of like powers of O, and obtain a sequence of differential equations and boundary conditions. We notice that the system of equations is uncoupled at each order. Indeed the equations of the potential at order m+1 depend on solutions of equilibrium equation at order m, so we can solve alternatively equations for the flow and equations corresponding to equilibrium equation. So, at each order, we firstly integrate the equation derived from (3.12) with respect to wn subjected to the boundary condition (3.13). Secondly we insert this value into (3.14) to get the expression for g n , which contains an unknown constant of the pressure hn . This constant will be determined by the condition of volume conservation (3.11). Furthermore these equations
43
WXUQWRKDYHVROXWLRQVDVOLQHDUFRPELQDWLRQRI/HJHQGUH¶VSRO\QRPLDOs. Then by expanding all quantities in terms of these polynomials as:
¦ a r P [ ,
w n r ,T
mt0
m n
m
f n (T )
¦D
mt0
P ([ )
m n 2m
cos T , we derive equations for anm r and D nm which are subjected to condition
with [
(3.11), enabling us to calculate hn . The solutions for the potential and the free surface [71] are given as:
P [ ª 27 º 9§ 1 1 · O « v r ,T , O 1 P1 [ ¨ 4 P [ O O 2 2 2 ¸ 3 » 2 8 8 r 2r 10r ¹ © ¬ 160r ¼
f T , O 1 O
k
2
O 4
3 P2 [ O(O 2 ) 16
O O2
Using a computer symbolic algebra package, we obtained the first 24 terms of the above solution series in (3.14) as well as the series for equatorial and polar radius of the bubble given respectively by Rmax
f (0, O ) and Rmin
§S · f ¨ , O ¸ . These power series solutions are ©2 ¹
valid for very small parameter value of Oas required by the implicit function theorem. 5. Hermite-Padé approximation technique The governing equations of the flow of a drop or bubble in a uniform flow in dimensionless form i.e. equations (3.11-3.14), may be rewritten as G O, M 0 where M
v, f , k . We have
reported that the implicit function theorem [77] ensures the existence and uniqueness of a solution provided that the linear operator L O of O
wG O ,M O is invertible in a neighborhood wM
0 . In general the operator L(O) has a discrete spectrum. As long as none of
its eigenvalues is zero, the operator is invertible; there exists therefore a unique solution of
44
G O,M 0 . Suppose that as O crosses a critical value Oc one or more eigenvalues of L(O) becomes zero. So the operator L(O) is no longer invertible, which is a condition for Oc to be a bifurcation point or a limit point, this last category contains the turning points. For further details reference is made to the work of Sattinger [2]. Another method for calculating bifurcation points can be done by analyzing the obtained solution series. Most perturbation series have a finite number of terms as the case considered here. This is due to combinatorial growth and the memory exhaustion of the machine. Using symbolic calculus codes, it is now possible to calculate a sufficient number of terms in the series. The aim of series analysis is to obtain from the first N coefficients, as accurately as possible, the radius of convergence namely the distance to the origin of the nearest singularity. We should mention that there exists a variety of methods elaborated for extracting estimates of the critical parameters from a finite number of series coefficients. The most commonly used methods are the ratio method, initially developed by Domb and Sykes and expanded by many authors [4,20], and semi-numerical approximant methods, such as Padé approximants [28], Hermite-Padé approximants [55, 82, 83], etc. For our part, we use a new form of differential approximants (see refs. [55,84]) which are a subclass of Hermite-Padé approximants. This method has proved to be a useful tool in many branches of physics and applied mathematics to extract the singularities of a function from its power series expansion, as well as evaluate it on its branch cuts. It consists of a high-order linear differential equation with polynomial coefficients that is satisfied approximately by the partial sum of a power series. 6. Results and discussion The bifurcation procedure above is employed to identify the nature and location of the nearest singularity limiting the convergence of the series (3.15), and if this singularity corresponds to 45
a bifurcation point. This will be first applied to the aspect ratio e(O )
Rmax , which as noted, Rmin
is the ratio of polar and equatorial radii of the drop. The series expansion of the aspect ratio in powers of Ois e(O )
¦ anOn
(3.16)
nt0
where the first 24 coefficients an are tabulated in Table 3.1. Note that a0
1.
n
an
n
an
n
an
n
an
1
028125
7
02284 102
13
02697 103
19
04217 104
2
03320 101
8
01532 102
14
01955 103
20
03138 104
3
02679 101
9
01059 102
15
01425 103
21
02341 104
4
09437 102
10
07408 103
16
01045 103
22
01751 104
5
05755 102
11
05242 103
17
07694 104
23
01313 104
6
03443 102
12
03745 103
18
05686 104
24
09869 105
Table 3.1: Coefficients an of the aspect ratio of the bubble in a uniform flow The convergence of the series (3.16) may be limited by a singularity Oc assumed to be multiplicative, hence e(O) takes the form (2.16).We observe that the pattern of signs appearing in the series e(O) is fixed, thus the singularity Oc lies on the positive axis [28]. The Hermite-Padé approximation procedure in Section 5 above was applied to the first few terms of the solution series of e(O) and reference pressure k(O).The results highlighted in Tables 3.2 and 3.3 below, show the rapid convergence of the singularity Oc together with its
46
corresponding critical exponent J c , with gradual increase in the number of series coefficients utilized in the approximants.
M
N
Oc
Jc
1
9
1.269498134
0.1326461567
2
12
1.271698428
0.1305393731
3
15
1.273698428
0.1256461567
4
18
1.273876133
0.1257548136
5
21
1.273898428
0.1257612346
6
24
1.273890627
0.1257642367
Table 3.2: Computations showing the procedure rapid convergence and bifurcation point of the aspect ratio e(O) M
N
Oc
Jc
1
9
1.268673456
0.204678943
2
12
1.273498931
0.2106798945
3
15
1.273567125
0.2153482314
4
18
1.273673489
0.2154567806
5
21
1.273694523
0.2155012347
6
24
1.273705894
0.2155452313
Table 3.3: Computations showing the procedure rapid convergence and bifurcation point of the reference pressure k(O) One can see in Table 3.2 and 3.3 that the direct calculation gives the estimation Oc | 1.2737. We observed that five digits are valid because they do not change. By similar calculations; we
47
1. 1
0.
0 e-
0
0.
0.
0.
0.
0.
0.
0.
0.
0.
1
-0.5
-1
O -1.5
-2
Figure 3.6. Plot of the function O O (e 1) It is possible to calculate the truncated series (3.16) defining e by a division procedure. This function is plotted in Figure 3.5a. This graph is not compatible with the expected singularity (2.16). A slice of the bifurcation diagram in the plane (O , e) is shown in Figure 3.5b. It represents the expected variation of aspect ratio e with the Weber number O. Indeed, there is a critical value Oc (a turning point) such that, for 0 d O Oc there are two solution branches (labeled I and II).
These solution branches occur due to the nonlinear nature of the governing equations (3.113.14). The previous analysis showed that the shape of the bubble is stable for low Weber number and becomes unstable, predicting breakup, at higher Weber number. This indicates a loss of stability of the equilibrium. In this situation, a subcritical bifurcation takes place as Ocrosses a critical value.
49
To confirm the results set out in the tables 3.2-3, the inverse function O O (e 1) is considered. Its series is given by inverting the series e(O) [28]. One can see in figure 3.6that this function has a horizontal tangent for the values (e - 1 = - 0.5850, O The maximum value of Ocorresponds to the value of Oc up to three digits, which confirms the existence of a bifurcation point (i.e. a turning point), see [86]. In what follows we will outline some significant results available from the literature providing critical Weber numbers at which a drop becomes unstable, and give a comparison with the one explained here. Hartunian and Sears [64] have studied experimentally and theoretically the flow of gas bubbles moving uniformly in different liquids. For bubbles moving in pure liquids, relatively non-viscous, they predicted the onset of instability for a critical Weber number of about 1.5876. The case of a single bubble rising with steady velocity was examined by Moore [66] who approximated the bubble shape to be an oblate spheroid. Furthermore, he showed that the aspect ratio is given to first order, by F Weber number Wec
1
9 We . He pointed out that there is a critical 64
3.74 beyond which the symmetric shape is impossible. Note that the
Weber number proposed by many authors is defined as We = 2O. By means of a virial technique, El Sawi [66] obtains an expression relating the Weber number and the aspect ratio. To first order in We, this relation agrees with the result derived by Moore. According to his study, the critical Weber number Wec indicating the beginning of path instability is obtained to be Wec
3.27 , close to that found by Hartunian and Sears.
Miksis et al. [67] have studied numerically the shape of a deformed bubble in a uniform flow, a problem very similar to the one treated here. However, they do not keep the volume
50
constant. They observed similar shapes but for greater values of the Weber numbers. They obtained a maximum Weber number We = 3.23; above which the solution fails to exist. Meiron, in [69], studied exactly the same question as in this paper. He used a collocation method in spherical co-ordinates for small values of the Weber number. He noticed difficulty in the convergence of his calculations for a value of the aspect ratio F less than 1.4. He plotted the shapes for 1.806 d 2We2 d 3.0492 . In our notation, these values correspond to 0.903 d O d 1.5246 . This latter value is greater than the singular value that we obtained. He
never observes the pinching of the bubble. All we can say is that, in the common range of parameters the shapes are somewhat different. Shankar [72] despite some disagreements with previous authors about explicit representations for bubble shapes, declared that he finds a maximum Weber number, Wec
3.239 which
compares well with the value deduced by Miksis et al. Ryskin and Leal [70] have studied numerically the bubble deformation in an axisymmetric straining flow through a viscous liquid. For large Reynolds number i.e. a weakly viscous liquid, they predicted that the critical Weber number for bubble breakup is between 2.7 and 2.8. In our notations, these values coincide with 1.35 and 1.4. We note that Frankel and Weis [68] obtained a critical Weber number much higher than those found by the previous estimations. All these values are not consistent with the one obtained in the present investigation, but close to that found by [71] through an improved Sykes-Domb method. The indicative studies mentioned are compared with others in table 3.4.
Oc
Hartunian
Moore
El Sawi
Miksis
Frankel
Shankar
Meiron
Ryskin
Sero-Guillaume
Our result
1.5876
1.87
1.635
1.615
5.33
1.6195
1.2
1.4
1.2572
1.2737
Table 3.4: Comparison of different critical Weber number with the available literature
51
7. Conclusions We have studied the shape of a bubble (or a drop of fluid) in the irrotational flow of a perfect fluid. The domain perturbation method was developed here for computing the ensuing quantities, i.e. the potential velocity field u, and the interface shape function f. We restrict our attention to steady bubble shapes which are axisymmetric about the direction of rise. We obtained an explicit expression for the potential u and the function f as a power series expansion in the Weber number which characterizes the flow. Using a computer algebra system, it is now possible to calculate a sufficient number of terms in the series, to study precisely the solution. The series solutions we obtained were valid up to a critical Weber number Oc beyond which no axisymmetric solution exists. Accordingly, we can formulate that as O varies, the shape of drop also varies, and when this parameter crosses the critical value Oc, the breakup of the drop occurs. From a qualitative point of view, the analysis mentioned above may explain the instability of bubble shapes above a certain critical Weber number. Otherwise there is an exchange of stability in the sense that axisymmetric shapes are stable for O
